Tits-type Alternative for Groups Acting on Toric Affine Varieties

Abstract

Abstract Given a toric affine algebraic variety $X$ and a collection of one-parameter unipotent subgroups $U_1,\ldots ,U_s$ of $\mathop {\textrm {Aut}}(X)$, which are normalized by the torus acting on $X$, we show that the group $G$ generated by $U_1,\ldots ,U_s$ verifies the following alternative of Tits type: either $G$ is a unipotent algebraic group or it contains a non-abelian free subgroup. We deduce that if $G$ is $2$-transitive on a $G$-orbit in $X$, then $G$ contains a non-abelian free subgroup and so is of exponential growth.